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I'm trying to work out what the sky looks like on an Earth-like moon, in particular the length of day and apparent size of the other celestial bodies: the gas giant's size, the gas giant's other moon's size, and the relative size of the star). I'm kind of stuck on the numbers here. Maths is my weak point, so I stole Artifexian's formulas and tried a simulation on Universe Sandbox a little while ago, but it wouldn't tell me the length of day on the moons. I was left having to use my brain, but all I got was an internal blue screen. Can anyone help me?

I'm gonna give all my numbers down below, but here's the setting and the result I want: it's a system with two tidally-locked Earth-like moons orbiting a massive Gas Giant.

  • How short can Moon A's orbit around its Gas Giant be? I.e., how short can its days be? I'm ideally aiming for 28 hours, but up to a week would be okay-ish (both sound extremely fast to me though. Jupiter's moons can go that fast, but they don't look very hospitable to me).
  • How can I have people from Moon A sometimes see Moon B from close enough that you can see its cities lit at night? Is it even possible?
  • How big do the gas giant and the star it orbits around look?

Detailed info:

So this gas giant (8.3 Jovian masses) orbits a K-type star of 0.8 solar masses. (Though a friend told me that it'd be easier if the gas giant orbited a star hotter than the Sun, maybe a class F of 1.4 solar masses I don't really care either way. I've put in the numbers for both).

At least two of these moons are habitable.

Moon A is tidally locked and fairly close to 1 Earth mass. I don't really care for the mass or situation of Moon B as long as it's habitable. Both are on inclined planes, just because it's more believable. Now, I'd also like our gas giant AND Moon B to be visible to the naked eye from Moon A's sky. The gas giant should appear to be at least the size of Earth's moon, and Moon B's surface should be visible too, because I'd like for people on Moon A to see the cities lit at night on Moon B. Due to the tidal locking, the gas giant would have its different phases during the day and appear full during the night.

(Maybe relevant?) Both moons would have been terraformed to be habitable for humans, though Moon A already had somewhat primitive but intelligent forms of life (most of which were killed during the terraformation process).


The numbers:

If K-type Star (all numbers are relative to our Sun unless indicated otherwise)

  • Star mass = 0.8 Sun masses
  • Diameter = 0.8477
  • Temperature = 0.8934 = 5156.7 Kelvin
  • Lifetime = 1.75
  • Habitability = 1.7155 AU
  • Goldilock zone = between 95% and 137% of 1.7155, so between 0.68 and 0.7 AU = between 10 474 and 541 429 km

If F-type Star

  • Star mass = 1.4 Sun masses

  • Star luminosity = 3.8416

  • Diameter = 1.28

  • Temperature = 1.185

  • Lifetime = 0.43

  • Habitability = 1.96

  • Goldilock zone = between 95% and 137% of 1.96, so between 1.862 and 2.6852 AU


Gas Giant

  • Gas Giant mass: 8.3 Jovian masses
  • Orbit: 0.68 AU from K-type star or 2.1 AU from F-type star? (I just went through the forum and learnt that tidal locking helps heat a moon, but I don't know how to account for that in the numbers)
  • Length of orbit: No idea. Sandbox Sim won't tell me, and whenever I accelerate time in the simulation it sends my planets flying off into deep space...
  • Velocity: 48.4 km/s (according to my simulation)

Moon A

  • Mass: 0.8 Earth mass
  • Orbit around Gas Giant: 23.8 hours in my simulation (which sounds AWEFULLY fast! Is that even possible?)
  • Velocity: 45.1 km/s
  • Semimajor axis: 58 007 km (periapsis 24000; apoapsis 92000)
  • Eccentricity: 0.59 (Inclination 77.44°; perihelion 178°; node 156°; mean 138°)

Moon B

  • Moon B mass: 1.4 Earth mass
  • Orbit around Gas Giant: 2.40 days in my simulation.
  • Velocity: 60.5 km/s
  • Semimajor axis: 1.05M km
  • Eccentricity: 0.061 (Inclination 0.40°; perihelion 305°; node 174°; mean 177°)

How many of these numbers seem off? I could send the Sandbox file if it helps, though I woudn't be surprised if the sim's rubbish too.

Many, many thanks! And sorry for the terribly messy post. (No wonder I kept sending my moons flying in Sandbox Simulator: I can't even keep my sentences in orbit.)

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    $\begingroup$ Your Semimajor axis for Moon A looks off by an order of magnitude. It should be more like 580 000 km. Otherwise, everything seems realistic. $\endgroup$ – Alexander Feb 4 '17 at 0:24
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    $\begingroup$ Ditto on the Semimajor Axis for Moon A, given that the radius of Jupiter is ~70,000km. Using this calculator (1728.org/kepler3a.htm) you can find the your missing orbital information. The Gas Giant's orbital period would be ~228 days around the K-type star, and ~937 days around the F-type. For Moon A to have 28 hour orbits its Semimajor axis should be 401,980km. $\endgroup$ – layagyasz Feb 4 '17 at 0:41
  • $\begingroup$ @S.E. & Alexander that would probably make a very helpful answer, but they can't "accept" a comment. $\endgroup$ – Zxyrra Feb 4 '17 at 15:17
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    $\begingroup$ The 77º inclination for moon A seems odd. I would expect all major moons in the same plane, and about the equatorial plane of the planet - specially since they are tidally locked. $\endgroup$ – Pere Feb 5 '17 at 21:22
  • $\begingroup$ Apparently the most common way for mooned gas giants to be found in the habitable zone is to migrate and capture nearby planets. This implies some degree of violence, hence the weird inclinations. I don't know the frequence of such events however, so I've no idea whether parallel planes would be more common. AFAIK though, for giants that migrate with their moons, the moons have a higher chance of being landless due to thawing. But it may be a preconception based on the assumption that the moons would have a similar amount of water as Earth. $\endgroup$ – Udon Nomaneim Feb 8 '17 at 15:09
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You list a semimajor axis of 58007 km. This almost certainly won't work, as semimajor axis is measured from the center of the body, and this gas giant will have a radius comparable to Jupiter. So Moon A will be inside the planet.

However, 580,000 km is the right semimajor axis for the given orbital period (23.8 hours) and the given masses (8.3 Jupiter masses and 0.8 Earth masses). So assuming that, you're going to be roughly 7 or 8 planetary radii away from the planet - so it will look absolutely huge in the sky. (Earth is about 220 Moon radii from the Moon, and about 220 Sun radii from the Sun, which is why they look about the same size from Earth.) There's nothing like this in human experience - even the Earth as seen from the Moon by the Apollo astronauts doesn't compare, it was only about 4x the apparent diameter (16x the apparent area) as the Moon is seen from Earth.

Since Moon A is tidally locked, the gas giant will always be visible on one side, and would be incredibly bright -- there would be darkness only when the Sun passed behind the gas giant. If the orbits were arranged properly (the moon orbit around the gas giant was 'face-on' as seen from the Sun) this would (unless I've made a mistake) never happen, and inhabitants of the 'gas-giant-wards' side would never experience true night.

As for Moon B, this will be roughly the same size in the sky as our Moon seen from Earth when they're distant, somewhat larger when they're close. (The maximum distance between the two moons will be around 2 million km, when they're at apoapsis on opposite sides; that's about 5x the distance to our Moon, and 1.4 Earth masses implies a radius slightly larger than Earth [or maybe somewhat more, if the composition of Moon B has less metal than Earth, like Mars or our Moon] so at least 4x our Moon's.

As for the star, I'm a bit lost here. A K-type star with 0.8 solar masses will have significantly less luminosity than the Sun, but you give a habitable zone farther out than the Sun's. It ought to be much closer -- stars' luminosity increases non-linearly with mass. Tau Ceti (G8.5) and 107 Piscium (K1) have about 80% solar mass but about 50% solar luminosity. https://en.wikipedia.org/wiki/107_Piscium

Your F-type star has a lifespan short enough to be an issue for evolving native sapients [maybe - after all we have only one example] but for terraforming it shouldn't matter.

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  • $\begingroup$ Great! Thanks for the detailed answer! This is really, really helping! The semimajor axis is indeed 580,000 km. I made a typo :) I'll do the math again for the K-type star. The habitable zone struck me as odd too, but I figured there must have been an obscure reason why it was so large. But maybe the calculations I used were only for certain types of stars. I'll check. I'm really glad that Moon B can be seen properly. And the gas giant would make a majestic dome on the horizon! Thanks again for everything! $\endgroup$ – Udon Nomaneim Feb 18 '17 at 23:15
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If you're ever looking for some inspiration for how to describe this from the perspective of a person on this moon I would highly recommend Farmer in the Sky by Robert Heinlein. It describes a colonist on Ganymede after it was terraformed. Heinlein was notorious for doing his homework when he wrote his books, so I wouldn't be surprised if his description of relative sizes is accurate for Ganymede.

It's also worth noting that even though your star puts out about 50% of the light as our sun, it wouldn't look any dimmer than daylight on earth. A sunny day on earth is about 111,000 lux, while a well-lit room is only about 1,000 lux. When the human eye gets more light than it can handle, it just shuts out the rest. So even if your moon gets significantly less than half the sunlight as earth, a bright sunny day will still look just as bright.

I know you were looking for a more math-based answer, but hopefully, this can help you put together a good picture of how everything would appear.

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  • $\begingroup$ Good idea! I keep looking at pictures of what Jupiter looks like from its moons, but it's mightily hard to get any sense of the real sizes involved... That book is next on my list :) Thanks a lot for pitching in! $\endgroup$ – Udon Nomaneim Feb 18 '17 at 23:22
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There is a tool that will allow you to see how sky would look and is called Space Engine. You correctly noticed that Goldilock would further away through tidal heating and reflected light from giant (and possibly radiation belt hitting it) however moon very close would very be much much hotter on close side and that would cause tidal winds and similar. Giant apparent brightness could be 20 times less than sun which might not seem like much but that would be 20.000 more than moon. Further moon would have much longer night day cycle and that would affect life in its own way.

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